Consider the random hyperbolic polynomial, f(x) = 1a1 coshx+···+np × an coshnx, in which n and p are integers such that n ≥ 2, p ≥ 0, and the coefficients ak(k = 1,2, . . . ,n) are independent, standard normally distributed random variables. If νnp is the mean number of real zeros of f(x), then we prove that νnp = π−1 logn+ O{(logn)1/2}.

In a recent paper [7] we analyzed a numerical algorithm for computing the number of real zeros of a polynomial system. The analysis relied on a condition number κ(f) for the input system f . In this paper we look at κ(f) as a random variable derived from imposing a probability measure on the space of polynomial systems and give bounds for both the tail P{κ(f) > a} and the expected value E(log κ...

In a recent paper [9] we analyzed a numerical algorithm for computing the number of real zeros of a polynomial system. The analysis relied on a condition number κ(f) for the input system f . In this paper we continue this analysis by looking at κ(f) as a random variable derived from imposing a probability measure on the space of polynomial systems. We give bounds for both the tail P{κ(f) > a} a...

This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments and we call those zeros that lie on these segments, real. Our main results give estimates for the number of real zeros as the weight goes to infinity. Mathematics Subject Classification (2010). Primary: 11F11, 11F30. Se...

We study the distribution of complex zeros of Gaussian harmonic polynomials with independent complex coefficients. The expected number of zeros is evaluated by applying a formula of independent interest for the expected absolute value of quadratic forms of Gaussian random variables.

The expected number of real projective roots orthogonally invariant random homogeneous polynomial systems is known to be equal the square root Bézout number. A similar result for multi-homogeneous systems, through a product orthogonal groups. In this note, those results are generalized certain families sparse with no invariance assumed.

the aim of this paper is to classify the finite simple groups with the number of zeros at most seven greater than the number of nonlinear irreducible characters in the character tables. we find that they are exactly a$_{5}$, l$_{2}(7)$ and a$_{6}$.

The expected number of zeros a random real polynomial degree $N$ asymptotically equals $\frac{2}{\pi}\log N$. On the other hand, average fraction trigonometric increasing converges to not $0$ but $1/\sqrt 3$. An roots system polynomials in several variables is equal mixed volume some ellipsoids depending on degrees polynomials. Comparing this formula with Theorem BKK we prove that phenomenon no...

We study so-called real zeros of holomorphic Hecke cusp forms, that is zeros on three geodesic segments on which the cusp form (or a multiple of it) takes real values. Ghosh and Sarnak, who were the first to study this problem, showed that existence of many such zeros follows if many short intervals contain numbers whose all prime factors belong to a certain subset of the primes. We prove new r...